1. Field of the Invention
The present invention relates to pipe structures of wind instruments.
The present application claims priority on Japanese Patent Application No. 2010-29311 (filing date: Feb. 12, 2010), the content of which is incorporated herein by reference.
2. Description of the Related Art
Various types of music synthesizer technologies simulating sound-producing mechanisms of acoustic instruments have been developed and disclosed in various documents such as Patent Document 1, namely Japanese Patent No. 2707913. Patent Document 1 discloses a music synthesizer device which simulates and reproduces resonance characteristics of a resonance pipe having a conical surface by way of a branch joint of two straight pipes.
FIGS. 1A-1C illustrate an approximation of resonance characteristics of a resonance pipe having a conical surface. FIG. 1A is a longitudinal sectional view of a resonance pipe 200 having a conical surface 204. The resonance pipe 200 is made of a hollow circular cone having a rotation axis X1 and a vertex V, which is truncated at a position of a distance R (measured from the vertex V) and at another position of a distance (R+L) in a direction indicated by an arrow D1. An opening 201 is formed at the position of the distance (R+L) from the vertex V, whilst another opening 202 is formed at the position of the distance R from the vertex V. S denotes a hollow area of the opening 202, and S2 denotes a hollow area of the opening 201. The area S differs from the area S2 in the resonance pipe 200. That is, the resonance pipe 200 is a tapered pipe having different sectional areas at opposite ends. In this connection, the rotation axis X1 refers to a rotation axis of a tapered pipe; the opening 201 having a large sectional area refers to a lower base; the opening 202 having a small sectional area refers to an upper base; a length L between the upper base and the lower base refers to a height; and a truncated length R refers to a distance between the vertex and the upper base.
An air column 203 inside the resonance pipe 200 resonates to sound input to the opening 202. Herein, c denotes a sound velocity of input sound; p denotes an air density of the air column 203; and k denotes the wave number of sound. In the case of a perfect reflection of sound at the opening 201 without considering attenuation due to friction of air inside the resonance pipe 200, an input acoustic impedance of the resonance pipe 200 viewed in the direction D1 is expressed by Equation (1).
                                                        Z              =                            ⁢                                                j                  ·                  ρ                  ·                  c                  ·                  k                  ·                  R                  ·                                      sin                    ⁡                                          (                                              k                        ·                        L                                            )                                                                                        S                  ⁢                                      {                                                                  sin                        ⁡                                                  (                                                      k                            ·                            L                                                    )                                                                    +                                              k                        ·                        R                        ·                                                  cos                          ⁡                                                      (                                                          k                              ·                              L                                                        )                                                                                                                }                                                                                                                          =                            ⁢                              1                                                      S                                          j                      ·                      ρ                      ·                      c                      ·                      k                      ·                      R                                                        +                                      S                                          j                      ·                      ρ                      ·                      c                      ·                                              tan                        ⁡                                                  (                                                      k                            ·                            L                                                    )                                                                                                                                                                            (        1        )            
Upon substituting counterpart terms of Equation (1) with Equations (2) and (3), it is possible to produce Equation (4).
                              Z          R                =                              j            ·            ρ            ·            c            ·            k            ·            R                    S                                    (        2        )                                          Z          L                =                              j            ·            ρ            ·            c            ·                          tan              ⁡                              (                                  k                  ·                  L                                )                                              S                                    (        3        )                                          1          Z                =                              1                          Z              R                                +                      1                          Z              L                                                          (        4        )            
Equation (4) shows that Z is produced via a parallel connection of ZR and ZL. Herein, ZR can be approximated to Equation (5) when kR is adequately small.
                              Z          R                =                                            j              ·              ρ              ·              c              ·              k              ·              R                        S                    ≈                                    j              ·              ρ              ·              c              ·                              tan                ⁡                                  (                                      k                    ·                    R                                    )                                                      S                                              (        5        )            
In Equation (5), ZL denotes an acoustic impedance of a straight pipe having the length L at an open end having the sectional area S. When kR is adequately small, ZR denotes an acoustic impedance of another straight pipe having the length R at an open end having the sectional area S. As described above, an acoustic impedance of the resonance pipe 200 is approximated by an acoustic impedance of the joint structure constituted of two straight pipes. In the following description, two pipes may approximate each other when they have similar acoustic impedances.
FIG. 1B is a longitudinal sectional view of a pipe unit 210 which approximates the resonance pipe 200. The pipe unit 210 is made of a hollow cylindrical pipe having a rotation axis X2, which are vertically cut at opposite positions. The pipe unit 210 has two openings 211 and 216, which are distanced from each other and positioned opposite to each other. Both the openings 211 and 216 have the same hollow area S. The same sectional area S is secured at any position of the pipe unit 210 perpendicular to the rotation axis X2. That is, the pipe unit 210 is a straight pipe whose sectional area is not varied at any position in the length direction. In this connection, the rotation axis X2 refers to a rotation axis of a straight pipe, and the distance between opposite openings refers to the length of a straight pipe.
Specifically, the pipe unit 210 has a joint structure constituted of a straight pipe 214 having a length L and another straight pipe 215 having a length R. The straight pipe 214 has the opening 211, whilst the straight pipe 215 has the opening 216. The same sectional area is secured in both of the straight pipes 214 and 215. In actuality, it is difficult to produce a completely straight pipe whose sectional area is not varied at any position in the length direction. Practically, pipes having very small variations of sectional areas within an allowable range of significant digits of Approximate Equation (5) can be assumed to be straight pipes. The following description is made on an assumption that the sectional area of each straight pipe is not practically varied.
The straight pipe 214 embraces an air column 213 therein. The air column 213 has the length L along the rotation axis X2 of the straight pipe 214. For the sake of convenience, the length of an air column inside a straight pipe is deemed equivalent to the length along the rotation axis of the straight pipe. In addition, the length of an air column inside a tapered pipe is deemed equivalent to the length along the rotation axis of the tapered pipe. Sound is input to a joint portion of the pipe unit 210 (indicated by an arrow D2) between the straight pipes 214 and 215. Equation (6) is created by applying a positive constant H to Equation (5).
                              Z          R                =                                            j              ·              ρ              ·              c              ·              k              ·              R                        S                    =                                                    j                ·                ρ                ·                c                ·                H                ·                k                ·                R                            HS                        ≈                                          j                ·                ρ                ·                c                ·                                  tan                  ⁡                                      (                                          k                      ·                      H                      ·                      R                                        )                                                              HS                                                          (        6        )            
Herein, kR is multiplied by H (which is adequately smaller than “1”) and converted into kHR so as to produce tan(kHR), thus improving an approximation precision. When kHR is adequately small, Equation (6) shows an acoustic impedance of a straight pipe having an open end with a sectional area HS and a length HR. This indicates an approximation of the resonance pipe 200 by use of two straight pipes having different thicknesses. FIG. 1C is a longitudinal sectional view of a pipe unit 220 which approximates the resonance pipe 200. The pipe unit 220 has a joint structure constituting of a straight pipe 224 having a sectional area S and a length L and a straight pipe 225 having a sectional area HS and a length HR. An air column 223 having the length L is formed inside the straight pipe 224. Sound is input to a joint portion of the pipe unit 220 (indicated by an arrow D2) between the straight pipes 224 and 225.
FIG. 2 is a graph showing impedance curves of pipe units. Herein, IC210 denotes an impedance curve of the pipe unit 210, and IC220 denotes an impedance curve of the pipe unit 220. As shown in FIG. 2, the pipe units 210, 220 differ from each other in terms of a degree of harmony (or consonance) at peak frequencies of the impedance curves IC210, IC220. Herein, the pipe unit 220 deviates in consonance more than the pipe unit 210; hence, the pipe unit 220 may approximate the property of a tapered pipe. Patent Document 1 discloses an approximation of the resonance pipe 200 by use of a straight pipe applied to an acoustic instrument.
FIG. 3A shows an example of a wind instrument 100 in which a mouthpiece 300 is attached to an input portion of the resonance pipe 200 having the conical surface 204. A cork member is attached to the input portion of the resonance pipe 200. The input portion of the resonance pipe 200 is inserted into the mouthpiece 300 via the cork member.
FIG. 3B shows another example of a wind instrument having a branch joint, which may serve as a saxophone. This wind instrument approximates the pipe structure of the wind instrument 100 shown in FIG. 3A in which the resonance pipe 200 extends from the inside of the mouthpiece 300. Specifically, a straight pipe 231 is inserted into and mouthpiece 300 such that an opening 800 (which runs through the straight pipe 231 and the mouthpiece 300) is formed at a joint portion therebetween, wherein an attachment 801 is engaged with the opening 800. The attachment 801 implements the functionality of the foregoing straight pipe having a length HR and a sectional area HS. For the sake of convenience, the straight pipe 231 refers to a main pipe; the attachment 801 refers to an auxiliary pipe; and a branch pipe is interposed between the main pipe and the auxiliary pipe. The auxiliary pipe differs from sound holes (which will be discussed later) whose open ends are opened or closed to produce a desired pitch of sound. In contrast, an open end of the auxiliary pipe is normally opened to produce a desired pitch of sound.
Since the auxiliary pipe is disposed at the position of the mouthpiece, a small hole needs to be pierced through the mouthpiece to communicate with the auxiliary pipe. This mechanism leads to a positional fixation of the mouthpiece, which prevents a player from replacing the mouthpiece with a preferred mouthpiece.